Department of Teaching and Learning, Steinhardt School of Culture, Education, & Human Development
Department of Mathematics, Courant Institute of Mathematical Sciences
New York University
MTHED-UE-1049: Mathematical Proof and Proving (MPP)
MATH-UA-125: Introduction to Mathematical Proofs
Instructors:
Prof. Jalal Shatah, Dept. of Mathematics
Prof. Orit Zaslavsky, Dept. of Teaching & Learning
Spring, 2012
Monday 12:00-2:00pm, Kimmel Center, room 804
Course Description
The course introduces elements of mathematical proof, focusing on three main themes: 1. The meaning
of mathematical statements – universal/existential; 2. The roles of examples in determining the validity
of mathematical statements; 3. The various forms and methods of mathematical proofs, including direct
(deductive) proof; proof by exhaustion; indirect proof (by contradiction, or by contrapositive);
mathematical induction; disproof by counterexample. This is a problem-based course. Lessons are
structured around activities that engage students in doing proofs that are meaningful to them and based
on mathematical topics with which they are familiar.
Course Objectives
Students should be able to:
 Apply various methods of proving to mathematical statements that are taken from
elementary number theory, geometry, basic combinatorics;
 Present proofs in an accurate and coherent way;
 Evaluate the validity of proofs (some with flaws).
Recommended Books (will be put on reserve in the library)
Chartrand, G., Polimeni, A. D., & Zhang, P. (2008). Mathematical proofs: A transition to
advanced mathematics. NY: Pearson, Addison Wesley.
Fendel, D., & Resek, D. (1990). Foundations of higher mathematics: Exploration and proof. NY: AddisonWesley Publishing Company.
Additional Resources
Jacobs, H. R. (1974). Geometry. NY: W. H. Freeman and Company. Chapter 1: The nature of deductive
reasoning (pp. 8-70).
Movshovitz-Hadar, N., & Webb, J. (1998). One Equals Zero and other mathematical surprises, paradoxes,
fallacies, and mind bogglers. Key Curriculum Press.
Nelson, R. B. (1993). Proofs without words. MAA.
2
Assignments
1. Weekly assignments:
On a weekly basis, students will receive an assignment that (usually) has two parts:
(i) A task requiring application of a proof method or principle that was illustrated and
discussed in the previous class;
(ii) A mathematical activity that prepares them for dealing with a method that will be
introduced in the following class.
Students will need to exhibit accurate and clear presentation of their claims.
2. Mid-term exam
3. Final exam
Assessment and Grading
Active participation in class activities and contribution to discussions and debates – 20%;
Weekly assignments – 40%;
Mid-term exam – 20%.
Final exam – 20%.
The grading of assignments and exams will be based on mathematical correctness and
coherence of presentations.
In order to get credit for the course, you must attend at least 10 (of the 12) classes, and submit on
time at least 80% of the weekly assignments. Late submissions may not be accepted. You also need to
take both mid-term and final exams.
Grading Rubrics for participation (out of 20 points):
A
Excellent (20 points)
B
Good (17 points)
C
Adequate (14 points)
Participates in class
activities and contributes to
discussions in every
session.
Participates in class
activities and contributes to
class discussions in most
sessions.
Participates in class
activities and contributes to
class discussions in some
sessions.
D
Poor (10 points)
Participates in class
activities but makes no
contributions or very few
contributions to class
discussions.
Note that points for unexcused absences may be deducted from the points in the above rubrics.
The final grade will be determined by the sum of points accumulated for each component
(participation, weekly assignments, and final project), according to Steinhardt School of
Education Grading Scale:
A
AB+
B
BC+
93-100
90-92
87-89
83-86
80-82
77-79
Syllabus for MPP course - Undergraduate Level
C
CD+
D
F
73-76
70-72
65-69
60-64
Below 60
October 2010January, 2012
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Course Outline (tentative)
Week
Topic
Week 1
Introduction to the course
Pre-test
Week 2
Symbolic Logic
Converting statements (such as "For every, …", "There exists,…") into mathematical notations,
thus making it clear what one has to prove;
The negation of mathematical statements;
The difference between logical implication and equivalence.
Week 3
The roles of examples in proof and proving
The roles of examples in determining the validity of mathematical statements, such as: one
supporting example is not sufficient for proving a valid universal statement while one supporting
example is sufficient for proving a valid existential statement; Additional supporting examples are
superfluous; One counter-example is sufficient for disproving a universal mathematical statement,
but not an existential statement.
Week 4
Direct Proof (part 1)
This is the most commonly used approach to proofs in mathematics.
The students will learn to prove statements such as "A implies B", or equivalently, "not B implies
not A", using inference rules (e.g., modus ponens and modus tollens).
Week 5
Direct Proof (part 2)
Direct proof applied to elementary number theory.
Week 6
Direct Proof (part 3)
Direct proof applied to geometry.
Week 7
Mid-term Exam (3/19/2012)
Proof by cases
Proof by exhaustion of all cases can be illustrated for several mathematical theorems, e.g., an
inscribed angle in a circle is half the measure of its corresponding central circle; two integers are
of the same parity if and only if their sum is even.
Week 8
Mathematical Induction (part 1)
Introduction to the principle of mathematical induction, and its equivalence to the principle of the
well ordered set of the Integers.
Concepts, definitions, notations, and conventions relates to mathematical induction.
Week 9
Mathematical Induction (part 2)
Mathematical induction applied to proving elementary properties of sums (finite and infinite).
Week 10
Mathematical Induction (part 3)
Mathematical induction applied to proving inequalities (e.g., Jensen's inequality for discrete sets).
Syllabus for MPP course - Undergraduate Level
October 2010January, 2012
4
Week
Week 11
Topic
Indirect Proof (part 1)
Proof by contradiction (i.e., assuming that a certain statement is false and from this assumption
arriving at a statement that contradicts some assumption made in the proof or some know fact) is
in some cases the only way to prove a statement (such as " 2 is an irrational number", or
Cantor's diagonalization procedure for proving that the numbers in the interval [0,1] are not
countable).
Week 12
Indirect Proof (part 2)
Proof by contrapositive (i.e., building on the logical equivalence between "A implies B" and "Not B
implies Not A") is often a useful method. To prove "If A then B" by the method of contrapositive
means to prove "If Not Q, Then Not P".
Week 13
Indirect Proof (part 3)
A direct approach to indirect proofs; e.g., the case of proving that there are an infinite number of
primes (Leron, 1985).
Week 14
Disproving
Disproving universal statements by generating counter-examples; Disproving existence
statements by indirect methods.
Summing up and reflections on the course
Note that throughout the course there will be opportunities to Evaluate Proofs. This part will help
students develop an understanding of the consequences of the chain of mathematical statements that
they make, i.e., what did they prove. We will discuss "faulty proofs" and learn to detect flaws in what
may seem as a convincing proof.
Syllabus for MPP course - Undergraduate Level
October 2010January, 2012
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Accommodation for NYU Students with Disabilities
Any student attending NYU who needs an accommodation due to a chronic, psychological,
visual, mobility, and/or learning disability, or is Deaf or Hard of Hearing should register with the
Moses Center for Students with Disabilities at 212 998-5980, 240 Greene Street. See
www.nyu.edu/csd.
Academic Integrity
The relationship between students and faculty is the keystone of the educational experience in
The Steinhardt School of Culture, Education, and Human Development at New York University.
This relationship takes an honor code for granted. Mutual trust, respect and responsibility are
foundational requirements. Thus, how you learn is as important as what you learn. A university
education aims not only to produce high quality scholars, but to also cultivate honorable
citizens.
Academic integrity is the guiding principle for all that you do; from taking exams, making oral
presentations to writing term papers. It requires that you recognize and acknowledge
information derived from others, and take credit only for ideas and work that are yours.
You violate the principle of academic integrity when you:
 Cheat on an exam;
 Submit the same work for two or more different courses without prior permission from
your professors;
 Receive help on a take-home examination that calls for independent work;
 Plagiarize.
Plagiarism, one of the gravest forms of academic dishonesty in university life, whether intended
or not, is academic fraud. In a community of scholars, whose members are teaching, learning
and discovering knowledge, plagiarism cannot be tolerated. Plagiarism is failure to properly
assign authorship to a paper, a document, an oral presentation, a musical score and/or other
materials, which are not your original work.
You plagiarize when, without proper attribution, you do any of the following:
 Copy verbatim from a book, an article or other media;
 Download documents from the Internet;
 Purchase documents;
 Report from other's oral work;
 Paraphrase or restate someone else's facts, analysis and/or conclusions;
 Copy directly from a classmate or allow a classmate to copy from you.
For more on academic integrity see http://steinhardt.nyu.edu/policies/academic_integrity.
Syllabus for MPP course - Undergraduate Level
October 2010January, 2012